Let f(1,y) = (r+ay°,r²y°,2r + by), g(u, v,w) = (3u² + bv² + w°, uvw). Find the derivative of f(g(u,v, w)) using the chain rule.

Q: Find the gradient of the function at the given point.

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Q: Use the Chain Rule to evaluate the partial derivative at the point specified. ∂h/∂q at (q, r) = (3,…

A: use the chain rule to evaluate the partial derivative dhdq at a point q,r=3,2  where hu,v=uev, u=q3,…

Q: If U = lx+my,v=mx-ly then prove that partial derivative of x

A: Given: u=lx+my v=mx-ly   To show: ux=-vy and uy=vx

Q: Find the gradient of the function at the given point f(x, y) = 5x + 4y² + 1, (1,4) Vf(1, 4) =

A: Solution of given question is written in below steps.

Q: Use the gradient to find the directional derivative of the function at P in the direction PQ. f(x,…

A:

Q: If f(x, y, z) 2x5 + 3y5 — 8x2у?z?, - what is the partial derivative of f with respect to z? .A f z =…

A: Partial differentiation of  F(x,y,z) = 2x⁵ +3y⁵ - 8x²y²z² with respect to z

Q: .    Find work done by the gradient of f(x, y, z) = zexy+ 2y2z  along the path from (1, 2, 4) to…

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Q: Find the linearization of the function f(x,y) = V9x² + 16y² at the point (1, 1), and use it to…

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Q: Use the Chain Rule to find at aw w = x² + y² + z²; x = 10tsin s, y = 10tcos s, z = 9st² əs Find the…

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Q: Use the Chain Rule to calculate the partial derivative. Express the answer in terms of the…

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Q: Use the limit definition of partial derivatives to find fx(x, y) and fy(x, y). f(x, y) = x2 − 4xy +…

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Q: The gradient of f(x,y,z)=xy2z3f(x,y,z)=xy2z3 at the point (1, 1, 1) is

A:

Q: Find the gradient of the function w = xy2z2,  and the maximum value of the directional derivative at…

A: Given function:    w=xy2z2

Q: Find the derivative of f at point P in the direction of u. f(x, y, z) = 6x² + 2y – 7z?, P = (8,–6,…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: (a) Find the derivative of ƒ(x, y, z) = x3 - xy2 - z at P0(1, 1, 0) in the direction of v = 2i - 3j…

A: We have given function fx,y,z=x3-xy2-z, point P01,1,0 and vector v=2i^-3j^+6k^

Q: Let f(x, y, 2) =x² + 3r°y? + 2°, g(u, v) = (u² + v², uv, 3u – 2v). Find the derivative of f(g(u, v))…

A:

Q: Find the gradient of the function z = ln(x2 − y) /(x − 4), at the given point     (2, 3)

A: To find the gradient differentiate z partially with  respect to x and y. Given function  z=lnx2-yx-4…

Q: f(x, Y, z) =

A: We are asked to find the gradient of the function.

Q: Use the Chain Rule to evaluate the partial derivative at the point specified. ag at (u, v) = (0, 1),…

A:

Q: Find the derivative of the function f (x, y, z) = -9x + 5y – 10z at the point (-3, 6, 5) in the…

A: we have to find the following . I have explained in details please see the explanation and if any…

Q: Q2 / Find the directional derivative of f(x.y.z) 4xy +5xz2 +6yz2 At the point P(3,1,4) in the…

A: ∂f∂x=4y+5z2∂f∂x=84∂f∂y=4x+6z2∂f∂y=108∂f∂z=10xz+12yz∂f∂z=168Therefore, the gradient is∇f (3, 1, 4)…

Q: Let f(x, y) = (x° + ay´, xʻy°, 2x+by), g(u,v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: f(x,y)=(x3+ay2,x2y3,2x+by)g(u,v,w)=(3u2+bv2+w3,uvw)To Find: Derivative of f(g(u,v,w)) using the…

Q: Find the gradient of f(x, y) =x2-5xy+y2 at the point (0, 1)

A: Gradient of the f(x,y) at a point (a,b) is given by :∇f=dfdx(a,b)i+dfdy(a,b)j

Q: Use the gradient to find the directional derivative of the function at P in the direction of  PQ.…

A: This is question of application of differentiation which we will proceed as follows

Q: Find the gradient of cos(xy) – 3y4 + 3 at the point (1,4) -

A:

Q: Find work done by the gradient of f(x, y, z) = zexy+ 2y2z  along the path from (1, 2, 4) to (-1, 0,…

A: The gradient of f(x, y, z) is:

Q: Find the directional derivatives of f(x,y,z) = x² y³ – y³z³ + x* in the direction of r = 2i– k, and…

A:

Q: f(x,y)=ln(5x+4y). Find the directional derivative of f at (1,0) in the direction of the line…

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Q: If f(x, y, z) = 2x5 + 3y5- 8x2y?z?, what is the partial derivative of f with %3D respect to z? fz =…

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Q: Find the direction derivative of f(x,y)= Xe ² + y ² at the paint (2,1), by the dirction of V = i- j

A: Topic:- application of derivatives

Q: Given f(x,y,z) = x2 + y2 + z2, find the maximum value of the directional derivative (df/ds) at the…

A: Consider the provided question,

Q: Compute the directional derivative of the function f(x, y, z) = 3xyz at the point P = (1, −3, −1) in…

A:

Q: Find the partial derivative of u=xy+yz+zx with respect to x.

A: Here, we need to find the partial derivative of u=xy+yz+zx with respect to x. So,              ∂ ∂…

Q: The gradient of f(x, y) xy at (x, y) = (– 1, 1) is

A:

Q: Find the directional derivative of F(x , y , z)=7y2e−x+3z2 in the direction u=(3,6,−2) at the point…

A: We have to solve

Q: Let f(x, y, z) = 3xy + 2yz - xz. Find the directional derivative of f(x, y, z) in the direction of…

A:

Q: a) If W = F(x ,y),where x =v²-g and y 2vg calculate W +W, %3D b) What is the directional derivative…

A: W=f(x,y) and x= v2-g2 & y=2vg   Wx = dW/dx = (df/dx*dx/dv) + (df/dx*dx/dg) ..............From…

Q: Find the gradient of f(x,y,z)=xy+yz+xz at point P(1,2,3).

A: As we know that the expression for the gradient is: ∇fx,y,z=i∂∂x+j∂∂y+k∂∂zfx,y,z Given:…

Q: Use the limit definition of the partial derivative to verify the formulas

A: By limit definition  of the partial derivative, ∂fx,y∂x=limh→0fx+h,y-fx,yh ,…

Q: Find the maximum rate of change of f(x, y, z) = tan(2x + 2y + 3z) at the point (−9, 6, 2) and the…

A:

Q: Find the maximum rate of change of f(x, y, z) = tan(2x + 7y + 5z) at the point (- 1, 1, 2).

A: We have to find

Q: Let f(r,y, z) = x² + 3x°y + z°, g(u, v) = (u² + v², uv, 3u – 2v). Find the derivative of f(g(u, v))…

A:

Q: Use the limit definition of partial derivatives to find fx(x, y) and fy(x, y). f(x, y) = x2 − 5xy +…

A: We need to find partial derivative.

Q: Find the gradient of the function f(x, y) = (x + y)/(y + 1),and the maximum value of the directional…

A: Given: fx,y=x+yy+1 The gradient of fx,y can be calculated as, ∇fx,y=i^∂∂xfx,y+j^∂∂yfx,y Here,…

Q: If f(x,y,z)=xe^(2y)sin(8z), then the gradient is ∇f(x,y,z)=

A:

Q: Use the gradient to find the directional derivative of the function at P in the direction of PQ.…

A: Given that, the function is gx,y,z=xye8z and the points are P3,12,0 and Q0,0,0.…